# Examples of homotopy Lie algebras

## Klaus Bering and Tom Lada

Address:

Institute for Theoretical Physics & Astrophysics Masaryk University, Kotlářská 2 611 37 Brno, Czech Republic

Department of Mathematics North Carolina State University Raleigh NC 27695

E-mail:

bering@physics.muni.cz

lada@math.ncsu.edu

Abstract: We look at two examples of homotopy Lie algebras (also known as $L_{\infty }$ algebras) in detail from two points of view. We will exhibit the algebraic point of view in which the generalized Jacobi expressions are verified by using degree arguments and combinatorics. A second approach using the nilpotency of Grassmann-odd differential operators $\Delta $ to verify the homotopy Lie data is shown to produce the same results.

AMSclassification: primary 18G55.

Keywords: homotopy Lie algebras, generalized Batalin-Vilkovisky algebras, Koszul brackets, higher antibrackets.